[FOM] 784: Revolutionary Possibilities/3
Harvey Friedman
hmflogic at gmail.com
Sun Jan 21 14:59:58 EST 2018
Incidentally, https://cs.nyu.edu/pipermail/fom/2018-January/020806.html
is an unnumbered posting that readers may be interested in. ALSO:
right now I will try to perfect theorems in elementary number theory
that use finite and infinite Ramsey theory, that
1. Almost certainly no one alive has any current chances of proving
them with finitistic methods or have any chance of obtaining bounds
with a fixed stack of exponentials.
2. We provocatively conjecture that no fixed stack of exponentials are
sufficient and in some cases we have independence from Peano
Arithmetic = PA.
3. For 1, I must initiate the appropriate conversations with top
number theorists for feedback.
This should follow the normal pattern: at first they are clumsy, and
then they gradually get more and more elegant, arriving at a very
satisfying perfectly natural form.
AND, after this, we will try to do this for ZFC by giving a series of
classical number theoretic interpretations of Emulation Theory,
starting with the clumsy towards the satisfyingly perfectly natural.
This will culminate in a web of applications of large cardinals to
elementary number theory, which cannot be handled by current number
theorists, with the statements accessible at the gifted high school
student level.
At the more advanced stages in both these PA and ZFC projects, I will
4. Seek to avoid a routine slavish translation of the combinatorial
into the elementary number theoretic. At some point, there should be
serious LEVERAGING of the great flexibility and power and simplicity
of elementary number theoretic concepts in order to greatly improve
the statements.
Should be call this RAMSEY NUMBER THEORY?
MY CURRENT FAVORITE
THIS INVOLVES ONLY THE EXISTENCE OF A SINGLE INTEGER RELATIVE TO ANY GIVEN ONE
THEOREM 1. For all k >= 1 there exists n >= 1 whose k maximal factors
have increasing (<=) quesidues.
A maximal factor of n is a factor of n that is less than n, which is
not a factor of any factor of n that is less than n.
The quesidue of positive integer n is the quadratic residue of n
modulo the least prime > n.
I prove Theorem 1 using more than PA = Peano Arithmetic. Is this
necessary? It may be for number theorists.
We now present some alternatives.
DISTINCT SUMS
THEOREM 2. Let n,m be positive integers. There is a set of m prime
numbers where all equal length distinct sums have the same quadratic
residue mod n.
This follows from the usual classical finite Ramsey theorem, 1930.
Not hard enough?
THEOREM 2'. Let n,m,r be positive integers. There is a set of m prime
numbers > n where all equal length distinct sums have the same r-th
power residue mod n.
Also from the usual classical finite Ramsey theorem, 1930.
To put more than PA = Peano Arithmetic to use here, we have a number
of possibilities. First of all, we can use the "relatively large"
concept going back to Paris and Harrington, 1977.
THEOREM 3. Let n be a positive integer. There is a set S of prime
numbers, |S| = min(S), where all equal length distinct sums have the
same quadratic residue mod n.
THEOREM 3'. Let n,r be positive integers. There is a set S of prime
numbers, |S| = min(S), where all equal length distinct sums have the
same r-th power residue mod n.
HOWEVER, we prefer to stick with sets of primes of given size as follows.
THEOREM 4. There are sets of primes of any given finite size where any
two equal length distinct sums have the same quadratic residue mod all
of the elements less than all of their summands.
THEOREM 4'. There are sets of primes of any given finite size where
any two equal length distinct sums have the same r-th power residue
mod all of the elements less than all of their summands.
************************************************************************
My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
https://www.youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ
This is the 784th in a series of self contained numbered
postings to FOM covering a wide range of topics in f.o.m. The list of
previous numbered postings #1-699 can be found at
http://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/
700: Large Cardinals and Continuations/14 8/1/16 11:01AM
701: Extending Functions/1 8/10/16 10:02AM
702: Large Cardinals and Continuations/15 8/22/16 9:22PM
703: Large Cardinals and Continuations/16 8/26/16 12:03AM
704: Large Cardinals and Continuations/17 8/31/16 12:55AM
705: Large Cardinals and Continuations/18 8/31/16 11:47PM
706: Second Incompleteness/1 7/5/16 2:03AM
707: Second Incompleteness/2 9/8/16 3:37PM
708: Second Incompleteness/3 9/11/16 10:33PM
709: Large Cardinals and Continuations/19 9/13/16 4:17AM
710: Large Cardinals and Continuations/20 9/14/16 1:27AM
700: Large Cardinals and Continuations/14 8/1/16 11:01AM
701: Extending Functions/1 8/10/16 10:02AM
702: Large Cardinals and Continuations/15 8/22/16 9:22PM
703: Large Cardinals and Continuations/16 8/26/16 12:03AM
704: Large Cardinals and Continuations/17 8/31/16 12:55AM
705: Large Cardinals and Continuations/18 8/31/16 11:47PM
706: Second Incompleteness/1 7/5/16 2:03AM
707: Second Incompleteness/2 9/8/16 3:37PM
708: Second Incompleteness/3 9/11/16 10:33PM
709: Large Cardinals and Continuations/19 9/13/16 4:17AM
710: Large Cardinals and Continuations/20 9/14/16 1:27AM
711: Large Cardinals and Continuations/21 9/18/16 10:42AM
712: PA Incompleteness/1 9/23/16 1:20AM
713: Foundations of Geometry/1 9/24/16 2:09PM
714: Foundations of Geometry/2 9/25/16 10:26PM
715: Foundations of Geometry/3 9/27/16 1:08AM
716: Foundations of Geometry/4 9/27/16 10:25PM
717: Foundations of Geometry/5 9/30/16 12:16AM
718: Foundations of Geometry/6 101/16 12:19PM
719: Large Cardinals and Emulations/22
720: Foundations of Geometry/7 10/2/16 1:59PM
721: Large Cardinals and Emulations//23 10/4/16 2:35AM
722: Large Cardinals and Emulations/24 10/616 1:59AM
723: Philosophical Geometry/8 10/816 1:47AM
724: Philosophical Geometry/9 10/10/16 9:36AM
725: Philosophical Geometry/10 10/14/16 10:16PM
726: Philosophical Geometry/11 Oct 17 16:04:26 EDT 2016
727: Large Cardinals and Emulations/25 10/20/16 1:37PM
728: Philosophical Geometry/12 10/24/16 3:35PM
729: Consistency of Mathematics/1 10/25/16 1:25PM
730: Consistency of Mathematics/2 11/17/16 9:50PM
731: Large Cardinals and Emulations/26 11/21/16 5:40PM
732: Large Cardinals and Emulations/27 11/28/16 1:31AM
733: Large Cardinals and Emulations/28 12/6/16 1AM
734: Large Cardinals and Emulations/29 12/8/16 2:53PM
735: Philosophical Geometry/13 12/19/16 4:24PM
736: Philosophical Geometry/14 12/20/16 12:43PM
737: Philosophical Geometry/15 12/22/16 3:24PM
738: Philosophical Geometry/16 12/27/16 6:54PM
739: Philosophical Geometry/17 1/2/17 11:50PM
740: Philosophy of Incompleteness/2 1/7/16 8:33AM
741: Philosophy of Incompleteness/3 1/7/16 1:18PM
742: Philosophy of Incompleteness/4 1/8/16 3:45AM
743: Philosophy of Incompleteness/5 1/9/16 2:32PM
744: Philosophy of Incompleteness/6 1/10/16 1/10/16 12:15AM
745: Philosophy of Incompleteness/7 1/11/16 12:40AM
746: Philosophy of Incompleteness/8 1/12/17 3:54PM
747: PA Incompleteness/2 2/3/17 12:07PM
748: Large Cardinals and Emulations/30 2/15/17 2:19AM
749: Large Cardinals and Emulations/31 2/15/17 2:19AM
750: Large Cardinals and Emulations/32 2/15/17 2:20AM
751: Large Cardinals and Emulations/33 2/17/17 12:52AM
752: Emulation Theory for Pure Math/1 3/14/17 12:57AM
753: Emulation Theory for Math Logic 3/10/17 2:17AM
754: Large Cardinals and Emulations/34 3/12/17 12:34AM
755: Large Cardinals and Emulations/35 3/12/17 12:33AM
756: Large Cardinals and Emulations/36 3/24/17 8:03AM
757: Large Cardinals and Emulations/37 3/27/17 2:39AM
758: Large Cardinals and Emulations/38 4/10/17 1:11AM
759: Large Cardinals and Emulations/39 4/10/17 1:11AM
760: Large Cardinals and Emulations/40 4/13/17 11:53PM
761: Large Cardinals and Emulations/41 4/15/17 4:54PM
762: Baby Emulation Theory/Expositional 4/17/17 1:23AM
763: Large Cardinals and Emulations/42 5/817 2:18AM
764: Large Cardinals and Emulations/43 5/11/17 12:26AM
765: Large Cardinals and Emulations/44 5/14/17 6:03PM
766: Large Cardinals and Emulations/45 7/2/17 1:22PM
767: Impossible Counting 1 9/2/17 8:28AM
768: Theory Completions 9/4/17 9:13PM
769: Complexity of Integers 1 9/7/17 12:30AM
770: Algorithmic Unsolvability 1 10/13/17 1:55PM
771: Algorithmic Unsolvability 2 10/18/17 10/15/17 10:14PM
772: Algorithmic Unsolvability 3 Oct 19 02:41:32 EDT 2017
773: Goedel's Second: Proofs/1 Dec 18 20:31:25 EST 2017
774: Goedel's Second: Proofs/2 Dec 18 20:36:04 EST 2017
775: Goedel's Second: Proofs/3 Dec 19 00:48:45 EST 2017
776: Logically Natural Examples 1 12/21 01:00:40 EST 2017
777: Goedel's Second: Proofs/4 12/28/17 8:02PM
778: Goedel's Second: Proofs/5 12/30/17 2:40AM
779: End of Year Claims 12/31/17 8:03PM
780: One Dimensional Incompleteness/1 1/4/18 1:14AM
781: One Dimensional Incompleteness/2 1/6/18 11:25PM
782: Revolutionary Possibilities/1 1/12/18 11:26AM
783: Revolutionary Possibilities/2 1/20/18 9:43PM
Harvey Friedman
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